A099225 -id:A099225 - cp's OEIS frontend

A057897 Numbers which can be written as m^k-k, with m, k > 1.

2, 5, 7, 12, 14, 23, 24, 27, 34, 47, 58, 61, 62, 77, 79, 98, 119, 121, 122, 142, 167, 194, 213, 223, 238, 248, 252, 254, 287, 322, 340, 359, 398, 439, 482, 503, 509, 527, 574, 621, 623, 674, 723, 726, 727, 782, 839, 898, 959, 997, 1014, 1019, 1022, 1087, 1154

Offset: 1

Henry Bottomley, Sep 26 2000

nonn

It may be that positive integers can be written as m^k-k (with m and k > 1) in at most one way [checked up to 10000].

All numbers < 10^16 of this form have a unique representation. The uniqueness question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x-y for x > y > 1 and b > a > 1, which appears to have no solutions. - T. D. Noe, Oct 06 2004

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000325, A008865, A024024, A024037, A024050, A057895, A057898, A057899.

Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A074981 (n such that there is no solution to Pillai's equation), A099226 (numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1).

nLim=1000; lst={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst] (* T. D. Noe, Oct 06 2004 *)

ok(n)={my(e=2); while(2^e <= n+e, if(ispower(n+e, e), return(1)); e++); 0} \\ Andrew Howroyd, Oct 20 2020

upto(lim)={my(p=logint(lim,2)); while(logint(lim+p+1,2)>p, p++); Vec(Set(concat(vector(p-1, e, e++; vector(sqrtnint(lim+e,e)-1, m, (m+1)^e-e)))))} \\ Andrew Howroyd, Oct 20 2020

A253913 Numbers of the form m^k + m, with m >= 0 and k > 1.

Original entry on oeis.org

0, 2, 6, 10, 12, 18, 20, 30, 34, 42, 56, 66, 68, 72, 84, 90, 110, 130, 132, 156, 182, 210, 222, 240, 246, 258, 260, 272, 306, 342, 350, 380, 420, 462, 506, 514, 520, 552, 600, 630, 650, 702, 732, 738, 756, 812, 870, 930, 992, 1010, 1026, 1028, 1056, 1122, 1190, 1260, 1302

Offset: 1

Alex Ratushnyak, Jan 18 2015

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A099225, A253914.

N:= 10000: # for terms <= N
S:= 0, 2:
for k from 2 to floor(log[2](N)) do
  for m from 2 do
    v := m^k+m; if v > N then break fi;
    S:= S, v;
od od:
sort(convert({S}, list)): # Robert Israel, Apr 28 2019, changed Jul 8 2021

max = 1000; Sort[Flatten[Table[m^k + m, {m, 2, Floor[Sqrt[max]]}, {k, 2, Floor[Log[m, max]]}]]] (* Alonso del Arte, Jan 18 2015 *)

def aupto(lim):
    xkx = set(x**k + x for k in range(2, lim.bit_length()) for x in range(int(lim**(1/k))+2))
    return sorted(filter(lambda t: t<=lim, xkx))
print(aupto(1500)) # Michael S. Branicky, Jul 08 2021

Changed to include 0 and 2 by Robert Israel, Jul 08 2021

A099226 Numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1.

Original entry on oeis.org

27, 248, 2194, 32763

Offset: 1

T. D. Noe, Oct 06 2004

No other terms < 10^15. The intersection of A057897 and A099225. The representation question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x+y for y > x > 1 and b > a > 1.

			27 = 25^2+2 = 32^5-5, 248 = 7^3+3 = 2^8-8, 2194 = 3^7+7 = 13^3-3 and 32763 = 181^2+2 = 8^5-5.

Cf. A074981 (n such that there is no solution to Pillai's equation).

nLim=40000; lst1={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst1, n]; k++ ], {m, 2, Sqrt[nLim]}]; lst2={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst2, n]; k++ ], {m, 2, Sqrt[nLim]}]; Intersection[lst1, lst2]

A099227 Primes of the form m^k+k, with m and k > 1.

Original entry on oeis.org

11, 37, 67, 83, 227, 443, 521, 1091, 1523, 2027, 3251, 4099, 6563, 6569, 9803, 10651, 11027, 12323, 13691, 15131, 17579, 21611, 29243, 32771, 32783, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203

Offset: 1

T. D. Noe, Oct 06 2004

nonn

It appears that primes of this form are much less common than primes of the form m^k-k (A099228).

As N increases, squares <= N outnumber all higher powers <= N by an increasingly wide margin, so the above observation is increasingly a consequence of the fact that primes of the form m^2 + 2 are less common than primes of the form m^2 - 2. Among numbers of these two forms, multiples of 3 make up 2/3 of the former, but none of the latter. - Jon E. Schoenfield, Jun 05 2021

Vincenzo Librandi and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Librandi)

Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A093324 (least k such that n^k+k is prime).

Cf. A099228.

nLim=200000; lst={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Select[Union[lst], PrimeQ]

list(lim)=my(v=List()); for(e=2,logint(lim\=1,2), forstep(n=3-e%2,sqrtnint(lim-e,e),2, my(t=n^e+e); if(isprime(t), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Jun 23 2023

A253914 Numbers that can be represented as both a^x + x and b^y + b, for some a, b, x, y > 1.

Original entry on oeis.org

6, 18, 20, 30, 66, 258, 260, 732, 1026, 3130, 4098, 4100, 16386, 19686, 46662, 65538, 65540, 65552, 262146, 531444, 823550, 1048578, 1048580, 4194306, 9765630, 14348910, 16777218, 16777220, 16777224, 67108866, 268435458, 268435460, 387420492, 387420498, 1073741826

Offset: 1

Alex Ratushnyak, Jan 18 2015

nonn

Intersection of A099225 and A253913.

Includes a^(a*b)+a = (a^b)^a+a for a,b > 1. - Robert Israel, Apr 28 2019

			a(1) = 6 = 2^2 + 2, in this case a = b = x = y = 2.
a(2) = 18 = 2^4 + 2 = 4^2 + 2.
a(8) = 732 = 3^6 + 3 = 9^3 + 3.

Cf. A099225, A253913, A099226.

A253916 Numbers that can be represented as both x^y + y and b^c + b + c, for some b, c, x, y > 1.

Original entry on oeis.org

264, 1334, 4108, 373323, 6436371, 387420507, 1099511627816

Offset: 1

Alex Ratushnyak, Jan 18 2015

Intersection of A099225 and A253775.

			264 is in the list since 264 = 2^8 + 8 and 264 = 4^4 + 4 + 4.
a(2) = 1334 = 11^3 + 3 = 36^2 + 36 + 2.

Cf. A099225, A253775, A253914, A253917.

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A057897 Numbers which can be written as m^k-k, with m, k > 1.

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A253913 Numbers of the form m^k + m, with m >= 0 and k > 1.

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A099226 Numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1.

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A099227 Primes of the form m^k+k, with m and k > 1.

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A253914 Numbers that can be represented as both a^x + x and b^y + b, for some a, b, x, y > 1.

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A253916 Numbers that can be represented as both x^y + y and b^c + b + c, for some b, c, x, y > 1.

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